- #1
Haorong Wu
- 415
- 90
- Homework Statement
- From Bernard Schutz's A first course in general relativity, exercise 19, in chapter 8.6:
Suppose a spherical body of uniform density ##\rho## and radius ##R## rotates ridigly about the ##x^3## axis with constant angular velocity ##\Omega##. Write down the components ##T^{0\nu}## in a Lorentz frame at rest with respect to the center of mass of the body, assuming ##\rho##, ##\Omega##, and ##R## are independent of time. For each component, work to the lowest nonvanishing order in ##\Omega R##.
- Relevant Equations
- For dust, ##T= \rho U \otimes U ##.
For a perfect fluid, ##T=(\rho + p) U \otimes U+p g^{-1}##
The answer with no details is given by
First, I considered a spherical shell because I thought the velocities at different radius ##r## will be different and hence the four-momentum will be different, as well.
Then, I writed down the linear momenta by $$\epsilon^{ijk} r_i p_j = L_k$$ with ##\vec r = R \cos \Omega t \hat x +R \sin \Omega t \hat y##, ##\vec L = L_3 \hat z=\frac {4 \pi R^3 \rho \Omega ^2 d R} 3 \hat z##. So $$P_x=\frac {4 \pi R^3 \rho \Omega ^2 d R \cos \Omega t} {3} \rm{~and}$$ $$P_y=- \frac {4 \pi R^3 \rho \Omega ^2 d R \sin \Omega t} {3}.$$
So the velocity is $$V_x=\frac R 3 \Omega^2 \cos \Omega t, V_y=- \frac R 3 \Omega^2 \sin \Omega t,$$ and the four-velocity will be given by ##U=(\gamma, \gamma v_x, \gamma v_y, 0)## with ##\gamma=1/\sqrt {1-v_x^2-v_y^2}##.
I am not sure what to do next. I tried to use the equations for dust that ##T^{00}=\rho U^0 U^0##, that would yield ##\rho \gamma ^2##. From the given answer, I see that I should really work in the MCRF with ##\gamma=1##. But in such a frame, ##U^i=0##. Then I would not derive the correct form of ##T^{0i}## in the answer.
Looking for some hints. Thanks.
##T^{00}=\rho, T^{01}=-\rho \Omega x^2, T^{02}=\rho \Omega x^1, T^{03}=0.## The components ##T^{ij}## are not fully determined by the given information, but they must be of order ##\rho v^i v^j##, i.e. of order ##\rho \Omega^2 R^2##.
First, I considered a spherical shell because I thought the velocities at different radius ##r## will be different and hence the four-momentum will be different, as well.
Then, I writed down the linear momenta by $$\epsilon^{ijk} r_i p_j = L_k$$ with ##\vec r = R \cos \Omega t \hat x +R \sin \Omega t \hat y##, ##\vec L = L_3 \hat z=\frac {4 \pi R^3 \rho \Omega ^2 d R} 3 \hat z##. So $$P_x=\frac {4 \pi R^3 \rho \Omega ^2 d R \cos \Omega t} {3} \rm{~and}$$ $$P_y=- \frac {4 \pi R^3 \rho \Omega ^2 d R \sin \Omega t} {3}.$$
So the velocity is $$V_x=\frac R 3 \Omega^2 \cos \Omega t, V_y=- \frac R 3 \Omega^2 \sin \Omega t,$$ and the four-velocity will be given by ##U=(\gamma, \gamma v_x, \gamma v_y, 0)## with ##\gamma=1/\sqrt {1-v_x^2-v_y^2}##.
I am not sure what to do next. I tried to use the equations for dust that ##T^{00}=\rho U^0 U^0##, that would yield ##\rho \gamma ^2##. From the given answer, I see that I should really work in the MCRF with ##\gamma=1##. But in such a frame, ##U^i=0##. Then I would not derive the correct form of ##T^{0i}## in the answer.
Looking for some hints. Thanks.